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10 Black Holes and Gravitational Collapse
10.11 Let’s do some rough order of magnitude energy conversion estimates. Using
a convenient reference estimate what fraction of the rest energy is liberated in
a typical atomic or molecular reaction? What of a nuclear reaction? If a mass
falls onto the surface of a neutron star estimate roughly what fraction of its
rest energy turns into kinetic energy and then into heat. What of a mass falling
into a black hole? (You may use a rough estimate for the potential energy of
a particle in the Schwarzschild metric using the classical potential.)
10.12 There is a heuristic motivation for taking the uncertainty in the position
of a particle near a black hole to be the Schwarzschild radius 2m. One may
calculate the electric field of a small electric charge near the black hole surface
in the Schwarzschild metric; it turns out that the field lines wrap around the
surface in such a way that at a large distance they appear to diverge from
the center of the hole rather than a point near the surface where the charge
actually is; this may be interpreted as an uncertainty in position. Study this
using the references Ruffini (1971) and Adler (1976, 2001).
10.13 Can you think of a heuristic way to show that the hawking radiation is
specifically thermal? (Nobody else has done this!)
10.14 The second heuristic way to estimate the Hawking temperature is to do a
heat engine gedanken (thought) experiment. It goes like this: fill a box with
thermal radiation from a hot heat reservoir far from a black hole; lower the box
to the surface of the black hole to run an engine and do work; at the surface,
taken to be a cold reservoir, release the radiation to the surface; pull the empty
box back to the hot reservoir and repeat. The ideal efficiency of such a heat
engine, using the second law of thermodynamics, gives a rough estimate for
the effective temperature of the black hole. Note that the necessary minimum
size of the box is important. See Ohanian (1994).
10.15 Calculate the Hawking temperature for a black hole of solar mass, as given
in the text.
10.16 Use the Stefan-Boltzmann law of radiation for a black body to calculate the
energy radiated by a black hole. Use that result to estimate the lifetime of a
black hole before it completely evaporates away. See also Chap. 19 on the
end stages of black hole evaporation.
10.17 Ponder for yourself the idea that the black hole entropy appears to reside on
a 2-surface; does it really violate any physical laws or intuition? It may be
interesting to read some of the papers on the holographic principle. See also
the comments on the Planck scale in Chap. 19.
10 Black Holes and Gravitational Collapse
10.11 Let’s do some rough order of magnitude energy conversion estimates. Using
a convenient reference estimate what fraction of the rest energy is liberated in
a typical atomic or molecular reaction? What of a nuclear reaction? If a mass
falls onto the surface of a neutron star estimate roughly what fraction of its
rest energy turns into kinetic energy and then into heat. What of a mass falling
into a black hole? (You may use a rough estimate for the potential energy of
a particle in the Schwarzschild metric using the classical potential.)
10.12 There is a heuristic motivation for taking the uncertainty in the position
of a particle near a black hole to be the Schwarzschild radius 2m. One may
calculate the electric field of a small electric charge near the black hole surface
in the Schwarzschild metric; it turns out that the field lines wrap around the
surface in such a way that at a large distance they appear to diverge from
the center of the hole rather than a point near the surface where the charge
actually is; this may be interpreted as an uncertainty in position. Study this
using the references Ruffini (1971) and Adler (1976, 2001).
10.13 Can you think of a heuristic way to show that the hawking radiation is
specifically thermal? (Nobody else has done this!)
10.14 The second heuristic way to estimate the Hawking temperature is to do a
heat engine gedanken (thought) experiment. It goes like this: fill a box with
thermal radiation from a hot heat reservoir far from a black hole; lower the box
to the surface of the black hole to run an engine and do work; at the surface,
taken to be a cold reservoir, release the radiation to the surface; pull the empty
box back to the hot reservoir and repeat. The ideal efficiency of such a heat
engine, using the second law of thermodynamics, gives a rough estimate for
the effective temperature of the black hole. Note that the necessary minimum
size of the box is important. See Ohanian (1994).
10.15 Calculate the Hawking temperature for a black hole of solar mass, as given
in the text.
10.16 Use the Stefan-Boltzmann law of radiation for a black body to calculate the
energy radiated by a black hole. Use that result to estimate the lifetime of a
black hole before it completely evaporates away. See also Chap. 19 on the
end stages of black hole evaporation.
10.17 Ponder for yourself the idea that the black hole entropy appears to reside on
a 2-surface; does it really violate any physical laws or intuition? It may be
interesting to read some of the papers on the holographic principle. See also
the comments on the Planck scale in Chap. 19.